Question 85129
1. A, B, and C can finish a job in 6 days. If B & C work together, the job will take 9 days; if A and C work together, the job will take 8 days. In how many days can each man working alone do the job?
:
I was confident that the all solutions would be integers, but it was not the case with c; Here's what I got, using 3 equations:
Let the completed job = 1
:
Eq1: {{{6/a + 6/b + 6/c}}} = 1
:
Eq2: {{{9/b + 9/c}}} = 1
:
Eq3: {{{8/a + 8/c}}} = 1
:
Find b in terms of c using eq2:
9c + 9b = bc; multiplied thru by bc
9b - bc = -9c
b(9-c) = -9c
b = {{{-9c/(9-c)}}}
:
Find a in terms of c using eq3:
8c + 8a = ac; multiplied thru by ac
8a - ac = -8c
a(8-c) = -8c
a = {{{-8c/(8-c)}}}
:
Substitute for a and b in eq1 (invert and multiply the dividing fraction):
6{{{((8-c)/(-8c))}}} + 6{{{((9-c)/(-9c))}}} + {{{6/c}}} = 1
:
{{{((48-6c)/(-8c))}}} + {{{((54-6c)/(-9c))}}} + {{{6/c}}} = 1
:
Multiply thru by 72c, get rid of the denominators and you have:
-9(48-6c) + -8(54-6c) + 72(6) = 72c
:
-432 + 54c - 432 + 48c + 432 = 72c
:
54c + 48c - 72c - 432 - 432 + 432 = 0 
:
30c - 432 = 
:
30c = +432
:
c = 432/30
:
c = 14.4 days required for c working alone
:
Find a using: a = {{{-8c/(8-c)}}}
a = {{{-8(14.4)/(8-(14.4))}}}
a = -115.2/-6.4
a = +18 days for a to do the job alone
:
Find b using:b = {{{-9c/(9-c)}}}
b = {{{-9(14.4)/(9-(14.4))}}}
b = -129.6/-5.4
b = +24 days for b
:
We should check the solutions in each original equation:
 eq1: {{{6/a + 6/b + 6/c}}} = 1
{{{6/18 + 6/24 + 6/14.4}}} = 1; I used a calc
:
Try checking solutions in eq2 and eq3
:
:
2. A bicyclist and a hiker leave at the same place at the same time and travel in the same direction. The bicyclist travel three times as fast as the hiker. at the end of 3 hr they are 24 miles apart. how fast does the hiker travel.
:
Let s = hikers speed in mph
Then 3s = bicyclist speed
:
Write a distance equation: dist = time * speed
bicyclist dist - hiker dist = 24 mi
3(3s) - 3s = 24
9s - 3s = 24
6s = 24
s = 4 mph is the hikers speed, obviously the bicyclist speed the 12 mph
:
check 3(12) - 3(4) = 24
:
:
3. A person traveling 4 hr by plane and 25 hr by ship covers 1580 miles. If the speed of the plane had been one-half of the actual speed and the speed of the ship had been one-fourth greater, the person would have traveled only 1315
miles in the same length of time. Find the speed of the plane and the ship.
:
let x = actual speed of the plane
let y = actual speed of the ship
:
Dist = time * speed:
actual: 4x + 25y = 1580
:
If equation:
.5(4x) + 1.25(25y) = 1315
2x + 31.25y = 1315
:
Mult above equation by 2 and subtract the actual equation:
4x +62.5y = 2630
4x +  25y = 1580
------------------subtracting eliminates x
0x + 37.5y = 1050
y = 1050/37.5
y = 28 mph, actual ship's speed
:
Find x using 2x + 31.25y = 1315
2x + 31.25(28) = 1315
2x + 875 = 1315
2x = 1315 - 875
x = 440/2
x = 220 mph, actual speed of the plane
:
Check solutions in the actual equation:
4(220) + 25(28) = 
880 + 700 = 1580